NIMBYs and their cousins, the Personal Owners of Public Parks, will use any ammunition they can get their hands on, no matter how big a stretch.
I don't know this particular park, but I doubt it's anything like the one where the lady was blinded.
On the other hand, soil damage is real---it's soil compaction, leading to loss of vegetation and erosion. As it stomping down the underbrush. If this park is actually a natural area, disc golf will impact that. Now, the Parks Dept. seemed to think is was a reasonable trade-off, so perhaps the opposition has inflated the issues. Wouldn't be the first time.
But, not knowing the park, it's also possible that disc golf is a poor fit for it. As much as we love disc golf, it doesn't belong everywhere.